Rigorous derivation of electromagnetic self-force

Abstract

During the past century, there has been considerable discussion and analysis of the motion of a point charge in an external electromagnetic field in special relativity, taking into account ``self-force'' effects due to the particle's own electromagnetic field. We analyze the issue of ``particle motion'' in classical electromagnetism in a rigorous and systematic way by considering a one-parameter family of solutions to the coupled Maxwell and matter equations corresponding to having a body whose charge-current density ${J}^{a}(\ensuremath{\lambda})$ and stress-energy tensor ${T}_{ab}(\ensuremath{\lambda})$ scale to zero size in an asymptotically self-similar manner about a worldline $\ensuremath{\gamma}$ as $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. In this limit, the charge, $q$, and total mass, $m$, of the body go to zero, and $q/m$ goes to a well-defined limit. The Maxwell field ${F}_{ab}(\ensuremath{\lambda})$ is assumed to be the retarded solution associated with ${J}^{a}(\ensuremath{\lambda})$ plus a homogeneous solution (the ``external field'') that varies smoothly with $\ensuremath{\lambda}$. We prove that the worldline $\ensuremath{\gamma}$ must be a solution to the Lorentz force equations of motion in the external field ${F}_{ab}(\ensuremath{\lambda}=0)$. We then obtain self-force, dipole forces, and spin force as first-order perturbative corrections to the center-of-mass motion of the body. We believe that this is the first rigorous derivation of the complete first-order correction to Lorentz force motion. We also address the issue of obtaining a self-consistent perturbative equation of motion associated with our perturbative result, and argue that the self-force equations of motion that have previously been written down in conjunction with the ``reduction of order'' procedure should provide accurate equations of motion for a sufficiently small charged body with negligible dipole moments and spin. (There is no corresponding justification for the non-reduced-order equations.) We restrict consideration in this paper to classical electrodynamics in flat spacetime, but there should be no difficulty in extending our results to the motion of a charged body in an arbitrary globally hyperbolic curved spacetime.